(a) (b)
(c)
Figure 6.8: Evolution ofRαLdl(a) restricted to the interior region and (b) restricted to the atmospheric
region, and (c) the entire region.
αL, along the fieldlines. Hence, a larger proportion of the twist is distributed to the atmospheric region in
Experiment2. The later drop off in the atmospheric helicity is partly due to the over-rotation of the field, but primarily due to the dissipation of the magnetic field.
6.3
Magnetic field strength comparison
Now that the effect of altering the length of domain is established, we seek to understand how a change in magnetic field strength affects this idealised model. By considering the model in this manner we separate the results of changing initial field strength from that of a change in length of fieldlines. In the experiments performed in Chapters4and5, a change in magnetic field strength allows the magnetic field to extend higher into the atmosphere, resulting in longer fieldlines. However, by considering this simple model we can understand which factor (field strength or fieldline length) is responsible for the difference in rotational velocities at the photosphere.
We use the same background stratification as in the length of box comparison and split this study into a comparison of two experiments as shown in Table6.3with the only difference being the initial field strength
6.3 Magnetic field strength comparison 162
(a) (b)
Figure 6.9: Visualisation of interior section of (a) the initial magnetic field and (b) the equivalent potential magnetic field.
(a) (b)
Figure 6.10: Evolution of the relative magnetic helicity,Hr, within (a) the interior volume and (b) the
atmospheric volume.
of the flux tube,B0. Note, theB0 = 5experiment is the same as Experiment1discussed in the previous
section with one small change. We had wished to keep these experiments the same but when we double
B0we find that the pressure deficit required for equilibrium exceeds the background gas pressure, resulting
in negative pressures. As this is unphysical, we need to triple the background gas pressure topb = 60.
Although this is only a problem in theB0 = 10case, we used the same background gas pressure in the
B0= 5experiment to make a closer comparison.
A comparison of the time evolution of rotation angles for theB0 = 5andB0 = 10experiments is
shown in Fig.6.11a. We note that in this case there is no emergence phase so the rotation angle begins to evolve att= 0. Clearly, a difference in the initial field strength alone has a dramatic impact on the evolution of the rotation angle but there is very little difference in the final angle of rotation. The photospheric rotation angle of theB0= 10tube drops more rapidly, experiencing an over-rotation before increasing to reach an
approximately constant rotation angle. This indicates the fieldlines threading through the sunspot rotate clockwise initially, causing an over-rotation, after which they begin to rotate anti-clockwise until a constant angle is reached. TheB0 = 5tube, on the other hand, behaves slightly differently. The initial change
6.3 Magnetic field strength comparison 163
Table 6.3: Magnetic field strength comparison set-up.
B0= 5 B0= 10
128×128×256gridpoints
−25< x <25,−25< y <25,−25< z <75
a= 2.5,R0= 15,α= 0.4
in rotation angle is both slower and smaller in magnitude, and the tube does not experience a definite over-rotation and subsequent rotation reversal.
(a) (b)
Figure 6.11: Comparison of the average rotation angle atz= 0as calculated using the method described in Section4.3.2for theB0= 5andB0= 10cases as coloured by the key. (a) shows the rotation angle over
time for the two cases and (b) shows the rotation angle over scaled time¯t=tB0for the two cases.
To compare the two cases on a more suitable timescale, we again redefine a scaled time¯t =tB0 to
taken into account the effect ofB0on the Alfv´en speed, shown in Fig.6.11b. This method for redefining
time is discussed in more detail in Section5.2. As we have doubledB0 from one case to the other, we
have also doubled the final scaled time. Even with this rescaling, there is still a discrepancy between the evolution of rotation angles. We note that the pressure within the tubes are different in the two cases, owing to the difference in pressure deficits as scaled byB2
0.
To analyse this comparison in more detail, we also present the evolution of the twist per unit length parameter,αLas a function of depth along the axis of the tube in Fig.6.12. It should be noted that each
of the twist per unit lengths,αL, are plotted at the same scaled time¯tto try and compare them at similar
stages in their evolution. The interior twist per unit length decreases by a larger amount and more rapidly in theB0 = 10case before increasing to match that of the corona. By the end of the experiment, the interior
and coronal twist per unit length match in theB0= 10case, indicating that a state of equilibrium has been
reached causing the flux tube to cease rotating. TheB0= 5case, on the other hand, is evolving on a slower
timescale and has not yet reached a constantαLalong the axis. We predict thatαLwill tend to a constant
value in theB0 = 5case and this will lead to the final agreement of rotation angles for theB0 = 5and
6.4 Summary 164
(a) (b)
(c) (d)
(e)
Figure 6.12: Variation ofαLas a function ofzalong the axis fieldline forB0= 5andB0= 10as coloured
by the key for scaled times (a)¯t= 0, (b)¯t= 100, (c)t¯= 200, (d)¯t= 500, and (e)t¯= 1000. We note that
¯
t=tB0is the scaled time with respect to the initial magnetic field strengthB0.
6.4
Summary
In this chapter we have conducted a series of simplified experiments to investigate the propagation of twist across an idealised interior-atmospheric region. In all cases, the set-up is split into a dense interior region and a rarefied atmosphere, and a vertical magnetic flux tube is defined such that all of the magnetic field’s
6.4 Summary 165
twist is contained within the interior connected to a straight untwisted atmospheric field. The flux tubes are set up in horizontal force balance and allowed to evolve. This results in an initial discontinuity in the twist per unit length,αL, and pressure,p, across the interior-atmosphere boundary. In all cases, torsional Alfv´en
waves are launched and a final state is reached in which the twist per unit length tends to a constant value along the axis fieldline. By considering domains of different lengths, we demonstrate the length of fieldlines are vital in determining the constantαLto which the system tends. Specifically, we find that the twist per
unit length is inversely dependent on the length of fieldlines. For example, if the twist per unit length tends to a value ofaalong a fieldline of lengthL, the twist per unit length would tend to a value ofa/2over a fieldline of length2L. As the rotation angle at the photosphere originates due to a twist imbalance, the rotation angle is dependent on the final twist per unit length. Hence, we also find larger rotation angles for longer fieldlines. This is consistent with our conclusions from Chapter5where we predicted that the fieldline length was crucial in determining the angle of rotation, and in turn the magnetic energy transported to the atmosphere.
We also investigate the impact of a change in the initial axial magnetic field strength on the evolution of twist and rotation angle, without altering the length of domain. By considering a change in magnetic field strength in this simplified set-up, we can separate the effects of changing the magnetic field strength from changing the length of fieldlines. These two effects are inherently linked in the earlier emergence experiments as stronger magnetic fields emerge more fully and their larger magnetic pressure allows the fields to expand higher into the atmosphere. In this case, we find that an increase in magnetic field strength changes the evolution of rotation angle, but not necessarily the final angle of rotation. In the small sample we have considered, we find that the tube in the strong field (B0 = 10) case experiences an over-rotation
while the weak field (B0 = 5) case rotates more slowly tends towards the final angle of rotation without
over-rotating. However, both cases appear to tend towards the same final angle of rotation. The final angle of rotation, does not necessarily give clues to the strength of field that lies under the photosphere, but rather the length of fieldlines threading through the region. However, as the two are inherently linked, longer fieldlines are most likely to originate from a strong sub-photospheric flux tube.
Chapter 7
Sunspot rotation due to
sub-photospheric velocities
In all previous chapters, we have considered single flux tubes that have been twisted prior to emergence. As we do not yet know the structure of the interior magnetic field, the addition of twist is merely an assumption based on simulations that predict distortion of untwisted magnetic fields on their rise through the interior. We have not yet investigated a case where the sub-photospheric flux tube is initially untwisted and how rotational convective velocities may influence the twist of the magnetic field, and the resulting rotation rate at the photosphere. A recent study bySyntelis et al.(2015) investigated the emergence of untwisted magnetic flux tubes and their subsequence expansion into the atmosphere accompanied by the onset of jets and heating of the plasma. However, this study did not consider the influence of sub-photospheric velocities, specifically the impact of these on the magnetic flux tube as it rises from below the photosphere.
The granular pattern of convection, described in Section1.1.1, can result in interesting convective ve- locities in the interior. The overturning flow at the edges of granules can produce horizontal vortices at the interface between the granule and intergranular lane (Nordlund et al.,2009). By taking the curl of the equation of momentum (Eq.1.10), it is clear that vorticity is generated by the cross product of gradients in density and pressure. Hence, vorticity is produced in locations where density and pressure gradients are not parallel, for instance at the mushroom heads of downdrafts (Nordlund et al.,2009). Downdrafts are sinks where cool plasma returns to the solar interior (Bonet et al.,2008). As the matter has angular momentum with respect to the draining point, it must spin up when nearing the sink, giving rise to a “bathtub” like whirl flow. Hence, we can compare these sub-photospheric vortices predicted by numerical simulations of convection to bath tub vortices (Bonet et al.,2008).
In order to incorporate these vortices into our model, we use rotational velocity drivers on the base of the domain. Velocity drivers on the boundary have been applied to magnetic fields in a variety of scenarios, mostly in coronal experiments where the driver is inserted on the photospheric boundary (e.g.Priest et al.,
2002,Wilmot-Smith,2015etc.). In the experiments in this chapter, we begin with an untwisted magnetic flux tube and allow it to emerge. However, at the same time we impose rotational velocity drivers on the
7.1 Standard case 168
footpoints of the flux tube on the lower boundary to inject twist into the flux tube. In all experiments, we impose a sub-photospheric velocity at a boundary4.25Mm below the solar surface. In order to study the effect of these rotations, we vary the size, magnitude, and number of velocity drivers, as well as the duration of driving.
The set-ups we choose are very highly idealised and it may seem overly simple to assume that the rotational drivers are at the footpoints of the flux tubes. However, asMeyer et al.(1979) andSchmidt et al.
(1985) noted, the tendency of emerging flux tubes to migrate towards the boundaries of super-granules and the congregation in the network is well established. Hence, it is reasonable to assume the footpoints of flux tubes could be caught in vortical motions at downdrafts.
The chapter is laid out as follows. In Section7.1, we describe the initial set-up of the standard case, with a strong magnetic flux tube and two fast, confined rotational velocity drivers that continue throughout the duration of the experiment. We outline the key results of this experiment before going on to discuss several cases in which we make one modification from the standard case. In Section7.2, we describe Case 1, in which we use a very similar set-up varying only the size of the rotational drivers, opting for a more spread driver surrounding the flux tube in its entirety. In Section7.3, Case 2 uses the same set-up as the standard case with two slower velocity drivers on the base. Similarly, in Case 3 we take the standard case and vary only the magnetic field strength of the flux tube, as described in Section7.4. Next, in Section7.5, we alter the number of drivers by considering only one driver on one of the footpoints. This experiment is, henceforth, referred to as Case 4. Finally, in Section7.6, Case 5 investigates an experiment in which we vary the driving time, only driving the experiment for the first75normalised time units. To conclude, in Section7.7, we briefly summarise the main findings of this chapter.
7.1
Standard case
The experiment begins with an untwisted magnetic flux tube (withα= 0in the initial magnetic field given in Eq.3.14). This results in a density excess of
ρexc=−B
2 0
2 e
−2r2/a2.
This is a larger density deficit than any non-zero twist values as the zero twist field has no inward magnetic tension force counteracting the outward magnetic pressure force. In this standard case, we set the axial field strength asB0= 10to model astrongmagnetic flux tube. The rest of the parameters are set up as before
witha = 2.5andR0 = 15. The axial field strength of the magnetic flux tube will be varied in Case 3
described later. A schematic of the initial set-up is shown in Fig.7.1with the same density and temperature stratification as that outlined in Chapter3. The only difference between the initial set-up of this experiment and those described in Chapters4and5is thatαis set to zero. This causes the fieldlines threading through the flux tube to appear straight, following the axis of the flux tube.
In order to twist the magnetic fieldlines and model a vortex at the centre of each footpoint, we subject the initial set-up to a driving velocity on the lower boundary of the domain. Specifically, we use spinning
7.1 Standard case 169
Figure 7.1: A 3D visualisation of the experiment with log profiles of the temperature on the back wall and density on the right wall as coloured by key on the left. A magnetogram of the vertical magnetic field is shown on the base of the domain as well as a select set of fieldlines shown in purple.
solid body footpoint motions described inDe Moortel and Galsgaard (2006) andWilmot-Smith and De Moortel(2007). In their experiments, they were modelling a photospheric rotation (like the sunspot rota- tions produced by emergence in earlier chapters) but the same general profile can be used lower down with a smaller driving speed. This is prescribed as theφcomponent of velocity (in a local(r, φ, z)cylindrical system) on the base of the domain:
vφ(r) =v0r1[1 + tanh(A(1−Br1))] +v0r2[1 + tanh(A(1−Br2))] =vφ1+vφ2 (7.1)
wherer2
1= (x−x1)2+ (y−y1)2andr22= (x−x2)2+ (y−y2)2such that(x1, y1)and(x2, y2)are the
centres of the footpoints. We have split the velocity intovφ1andvφ2, as given by the first and second terms
ofvφrespectively. This results in a Cartesian velocity driver of
vx = −vφ1sin(φ1)−vφ2sin(φ2),
vy = vφ1cos(φ1) +vφ2cos(φ2),
whereφ1= arctan ((y−y1)/(x−x1))andφ2= arctan ((y−y2)/(x−x2)). The coefficientv0affects
the magnitude and direction of the driving speed, andAandB describe the steepness and location of the drop off inv outside the sources, respectively. Note the velocity profile is designed to increase linearly with radial distance from the centre of each of the sources, in order to maintain the shape of the flux concentrations as they are rotated. We choose this type ofspinningrotation as it is localised only affecting the footpoint it is concentrated on. Other studies, such asDe Moortel and Galsgaard (2006), consider rotationalvelocity drivers that rotate multiple flux tubes with a single larger driver. In all experiments we set
Aas16.8and varyv0andBthroughout the experiments. We set(x1, y1) = (0,−15)and(x2, y2) = (0,15)
as these are the centres of the footpoints of the flux tube (as outlined in Chapter3). In the standard case, we setv0= 0.05for a counter-clockwise rotation andB = 0.4which we refer to as afast,confinedrotation.
7.1 Standard case 170
To demonstrate the velocity drivers, a coloured contour of vorticityωz= ∂vy ∂x − ∂vx ∂y is shown at
z =−25in Fig.7.2a. The positive sign of vorticity signifies a counter-clockwise rotation, and the black contours overplotted show the magnetic field concentrations. It is important to note in this case that the magnetic field contours surround the region of rotation and both footpoints are rotated. In Fig.7.2b, we have plottedvφ againstrcalculated on the base of the simulation domain. This shows that the velocity
increases linearly with radius from zero atr= 0untilr=a= 2.5where it drops off with atanhprofile. The velocity reaches a maximum of0.2or1.2km/s in physical units.
(a) (b)
Figure 7.2: Initial set-up of standard case with (a) thez-component of vorticity on base of simulation domain and (b) theφcomponent of velocity against radiusron the base of simulation domain.
To start the rotation smoothly, the driver is built up with atanhprofile:
vφ(r, t) =vφ(r) 1 2 1 + tanh t−2 2 , (7.2)
usingvφ(r)prescribed in Eq.7.1. The purpose of this is to increase the velocity gradually, in an attempt
to reduce shocks due to a sudden onset of velocity. The experiment is performed for150normalised time units, or62.5minutes. In order to work out the amount of twist injected, we use the relationvφ =r
dφ
dt
to work out the expected angle of rotation on the base. Let us focus on the footpoint subscripted1. For
r1<2.5,vφ≈2v0r1. Hence,
dφ
dt = vφ
r1